Group #15756684 — subgroups

Subgroups — #15756684

Order 3328.

382 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114988	C₂	1	1	yes	maximal, characteristic
2	#13114582	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718071	C₂	1	2	yes	maximal
4	C₂×#11718072	C₂	1	2	yes	maximal
5	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	—	2	2	no	—
10	C₂×C₄⋊#1055	C₂²	1	2	yes	—
11	C₂×C₄×#1055	C₂²	1	2	yes	—
12	C₁₃×#336	—	2	2	no	—
13	#11718116	C₄	1	2	yes	—
14	#11718116	C₄	1	2	yes	—
15	#11718072	—	2	4	no	—
16	#11718071	—	2	4	no	—
17	C₂²×C₈×C₁₃	—	2	1	no	—
18	C₂²×C₈×C₁₃	—	2	1	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	2	yes	—
22	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×#1055	—	2	2	no	—
25	C₂²×#1055	C₂×C₄	1	4	yes	—
26	C₁₃×C₄⋊C₈	—	2	2	no	—
27	C₁₃×C₄⋊C₈	—	2	2	no	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄⋊#1055	—	4	4	no	—
31	C₄×#1055	D₄	1	4	yes	—
32	C₄×#1055	Q₈	1	4	yes	—
33	#8812	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₃	—	4	1	no	—
40	C₂×C₈×C₁₃	—	4	1	no	—
41	C₂×C₈×C₁₃	—	4	2	no	—
42	C₂×C₈×C₁₃	—	4	2	no	—
43	C₄²×C₁₃	—	2	4	no	—
44	C₄²×C₁₃	—	2	2	no	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	—	2	2	no	—
47	C₄²×C₁₃	—	2	2	no	—
48	C₄²×C₁₃	C₂×D₄	1	2	yes	—
49	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
50	C₂×#1055	—	2	4	no	—
51	C₂×#1055	—	2	4	no	—
52	C₂×#1055	—	4	2	no	—
53	C₂×#1055	—	4	4	no	—
54	C₂×#1055	C₄⋊C₄	1	8	yes	—
55	#1842	—	13	1	no	—
56	#1831	D₁₃	1	1	yes	characteristic
57	#1744	—	13	1	no	—
58	C₂×#335	—	13	2	no	—
59	C₂×#334	—	13	2	no	—
60	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	#108	1	4	yes	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#131	1	2	yes	—
73	C₂×C₄×C₁₃	#131	1	2	yes	—
74	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₁₃	#137	1	2	yes	—
76	C₈×C₁₃	—	8	1	no	—
77	C₈×C₁₃	—	8	1	no	—
78	#1055	—	2	8	no	—
79	#1055	—	8	2	no	—
80	C₂×C₄⋊C₈	—	13	2	no	—
81	C₂×C₄⋊C₈	—	26	2	no	—
82	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
83	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
84	C₂×C₄×C₈	—	13	2	no	—
85	C₄³	C₂×D₁₃	1	1	yes	characteristic
86	#336	—	2	2	no	—
87	#336	—	13	2	no	—
88	#336	—	13	2	no	—
89	#335	—	26	4	no	—
90	#334	—	26	4	no	—
91	C₂²×C₁₃	#384	1	1	yes	characteristic
92	C₂²×C₁₃	#424	1	1	yes	characteristic
93	C₂²×C₁₃	#424	1	1	yes	characteristic
94	C₂²×C₁₃	#428	1	2	yes	—
95	C₂²×C₁₃	C₂×#108	1	2	yes	—
96	C₄×C₁₃	—	2	4	no	—
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	—	4	2	no	—
99	C₄×C₁₃	#424	1	4	yes	—
100	C₄×C₁₃	#496	1	1	yes	characteristic
101	C₄×C₁₃	#497	1	1	yes	characteristic
102	C₄×C₁₃	#501	1	2	yes	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	13	4	no	—
106	C₂²×C₈	—	26	2	no	—
107	C₂×C₄²	Dic₂₆	1	2	yes	—
108	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
109	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
110	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
111	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	26	2	no	—
115	C₄⋊C₈	—	26	2	no	—
116	C₄⋊C₈	—	52	4	no	—
117	C₄×C₈	—	13	4	no	—
118	C₄×C₈	—	13	4	no	—
119	C₂×C₁₃	#1573	1	2	yes	—
120	C₂×C₁₃	#1574	1	2	yes	—
121	C₂×C₁₃	#1843	1	1	yes	characteristic
122	C₂×C₁₃	#1847	1	1	yes	characteristic
123	C₂×C₁₃	#1848	1	1	yes	characteristic
124	C₂²×C₄	#6101	1	2	yes	—
125	C₂²×C₄	#6101	1	2	yes	—
126	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	13	8	no	—
134	C₂×C₈	—	26	4	no	—
135	C₂×C₈	—	26	4	no	—
136	C₂×C₈	—	52	2	no	—
137	C₂×C₈	—	52	4	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	4	no	—
143	C₄²	D₄×D₁₃	1	2	yes	—
144	C₄²	Q₈×D₁₃	1	2	yes	—
145	C₁₃	#8812	1	1	yes	characteristic, complemented, Sylow
146	C₂³	C₂×#6101	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#88062	1	4	yes	—
154	C₂×C₄	#88142	1	2	yes	—
155	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
156	C₂×C₄	#88155	1	2	yes	—
157	C₂×C₄	#88161	1	2	yes	—
158	C₂×C₄	#88194	1	1	yes	characteristic
159	C₂×C₄	#88198	1	1	yes	characteristic
160	C₂×C₄	#88203	1	2	yes	—
161	C₂×C₄	#88205	1	2	yes	—
162	C₈	—	8	1	no	—
163	C₈	—	8	1	no	—
164	C₈	—	26	8	no	—
165	C₈	—	104	2	no	—
166	C₂²	#11718637	1	1	yes	characteristic
167	C₂²	#11718656	1	2	yes	—
168	C₂²	#11718763	1	1	yes	characteristic
169	C₂²	#11718780	1	1	yes	characteristic
170	C₂²	C₂×#88062	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#11718500	1	4	yes	—
175	C₄	#11718719	1	1	yes	characteristic
176	C₄	#11718721	1	1	yes	characteristic
177	C₄	#11718726	1	2	yes	—
178	C₂	#13113622	1	2	yes	—
179	C₂	#13113624	1	2	yes	—
180	C₂	#13114989	1	1	yes	characteristic
181	C₂	#13115005	1	1	yes	characteristic
182	C₂	#13115007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114988

C₂

yes

maximal, characteristic

#13114582

C₂

yes

maximal, characteristic

C₂×#11718071

C₂

yes

maximal

C₂×#11718072

C₂

yes

maximal

C₁₃×#1831

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₈

C₂²

yes

characteristic

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

C₁₃×#336

—

#11718116

C₄

yes

—

#11718116

C₄

yes

—

#11718072

—

#11718071

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₂²×#1055

C₂×C₄

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

—

—

—

D₄

yes

—

C₄×#1055

Q₈

yes

—

#8812

—

maximal, Sylow

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

#1842

—

#1831

D₁₃

yes

—

—

—

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#108

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

—

C₄³

C₂×D₁₃

yes

—

—

—

—

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

C₂×#108

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#424

yes

—

100

C₄×C₁₃

#496

yes

characteristic

101

C₄×C₁₃

#497

yes

characteristic

102

C₄×C₁₃

#501

yes

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄²

Dic₂₆

yes

—

108

C₂×C₄²

C₄×D₁₃

yes

characteristic

109

C₂×C₄²

C₄×D₁₃

yes

characteristic

110

C₂×C₄²

C₂²⋊D₁₃

yes

—

111

C₂×C₄²

C₂²×D₁₃

yes

characteristic

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₂×C₁₃

#1573

yes

—

120

C₂×C₁₃

#1574

yes

—

121

C₂×C₁₃

#1843

yes

characteristic

122

C₂×C₁₃

#1847

yes

characteristic

123

C₂×C₁₃

#1848

yes

characteristic

124

C₂²×C₄

#6101

yes

—

125

C₂²×C₄

#6101

yes

—

126

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

127

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

128

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

D₄×D₁₃

yes

—

144

C₄²

Q₈×D₁₃

yes

—

145

C₁₃

#8812

yes

characteristic, complemented, Sylow

146

C₂³

C₂×#6101

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#88062

yes

—

154

C₂×C₄

#88142

yes

—

155

C₂×C₄

C₄⋊C₄×D₁₃

yes

—

156

C₂×C₄

#88155

yes

—

157

C₂×C₄

#88161

yes

—

158

C₂×C₄

#88194

yes

characteristic

159

C₂×C₄

#88198

yes

characteristic

160

C₂×C₄

#88203

yes

—

161

C₂×C₄

#88205

yes

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

104

—

166

C₂²

#11718637

yes

characteristic

167

C₂²

#11718656

yes

—

168

C₂²

#11718763

yes

characteristic

169

C₂²

#11718780

yes

characteristic

170

C₂²

C₂×#88062

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#11718500

yes

—

175

C₄

#11718719

yes

characteristic

176

C₄

#11718721

yes

characteristic

177

C₄

#11718726

yes

—

178

C₂

#13113622

yes

—

179

C₂

#13113624

yes

—

180

C₂

#13114989

yes

characteristic

181

C₂

#13115005

yes

characteristic

182

C₂

#13115007

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#8812
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₁₃

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1831

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
145. C₁₃	#8812	semidirect

H ≅ G/N

type

145. C₁₃

#8812

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13114988	2
#13114582	2
C₂×#11718071	2
C₂×#11718072	2
C₁₃×#1831	2
#8812	13

subgroup

index

C₁₃×#1831

Subgroups — #15756684

Conjugacy classes of subgroups

Complements

Maximal subgroups